Question

Tangents are drawn to the circle $x^2+y^2=50$ from a point $P$ lying on the $x-$axis. These tangents meet the $y-$axis at points $P_1$ and $P_2$. Possible coordinates of $P$ so that area of $△P{P}_1{P}_2$ is minimum, are

• A. $(10,0)$
• B. $(10\sqrt{2},0)$
• C. $(-10,0)$
• D. $(-10\sqrt{2},0)$

Answer 1

Answer: (A), (C)

The correct options are
A $(10,0)$
C $(-10,0)$


$OP=r\sec \theta =5\sqrt{2}\sec \theta$
Similarly, $O{P}_1=5\sqrt{2}\text{cosec}\theta$
$\therefore ar\left(△P{P}_1{P}_2\right)=\frac{100}{\sin 2\theta }$
$\left(ar\right(△P{P}_1{P}_2){)}_{min}=100$ at $\theta =\frac{\pi }{4}$
$\Rightarrow OP=10$
$\Rightarrow P=(10,0),(-10,0)$

Alternate Solution :

Area of $△P{P}_1{P}_2$ is minimum when the tangents $P{P}_1$ and $P{P}_2$ are perpendicular, so point $P$ will lie on the director circle of $x^2+y^2=50.$

As $P$ lies on the $x-$axis, hence coordinates of $P$ are given by
$(\sqrt{2}\times 5\sqrt{2},0)$ or $(-\sqrt{2}\times 5\sqrt{2},0)$
$\Rightarrow (10,0)$ or $(-10,0)$